Physics is the realm of rules and predictions. With a minimal set of parameters as input, physical laws can forecast the future, deduce the past, and unravel nature’s inevitable course. In such a world, the concept of chaos might seem misplaced. Yet, chaos theory is one of the most influential mathematical frameworks [1], and it has surprising real-world applications. Beyond science, chaos has also fascinated the general public: references to the so-called “butterfly effect” can be found all over pop culture [2, 3]. But what is chaos? And how do the rigid laws of physics reconcile with its apparently erratic nature?

MIT, 1961. On a winter day, meteorologist and mathematician Edward Lorenz decides to repeat one of his weather simulations. He takes a smart and seemingly innocent shortcut: instead of repeating the simulation from the start, he resumes it from the middle, copying the output from the previous one. This run, with the same inputs, should provide the same results. He types the numbers in his computer, clicks on “START” and goes for a coffee. When he comes back and looks at the results, he is puzzled: the output dramatically diverges from the expected one. After ruling out the possibility of a computer error or a flaw in his model, Lorenz realizes that every iteration of the same simulation produces an unpredictable, apparently random, result. [4].

Lorenz stumbled across one of the hallmarks of chaotic systems: sensitivity to initial conditions [5]. An infinitesimal error in the input numbers set off a completely different and unpredictable evolution of the simulated system. Originally termed “the sea-gull effect,” Lorenz later adopted the more poetic “butterfly effect” to describe this instability. Yet, somewhat counterintuitively, chaotic systems are still deterministic: simple, non-stochastic equations govern them, but their extreme sensitivity to inputs makes them practically unpredictable.

Following the evolution of a chaotic system in phase space might give you dizziness – irregular paths that never repeat themselves, always different depending on the starting point. But taking a step back and observing the global behavior of many of them together, chaos theorists found that they too follow recursive, periodical structures [5,6]. This paradoxical discovery reveals a hidden order within chaos, suggesting that even in the seemingly erratic dance of chaotic systems, there exists a larger, coherent organization. At the center of chaos theory is the fascinating idea that order and chaos are not always opposed.

The connection between chaos and order has helped scientists to model some of the most complex systems around us, leading to significant advances in weather forecasting [5], neurology [6], ecology [7], and interplanetary space travel [8, 9], to name just a few areas. Hence, chaos theory not only reveals an inner order in unpredictability, but also offers a valuable framework for understanding nature across various disciplines.